# Full text of "Treatise On Analysis Vol-Ii"

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```21    PRODUCT OF MEASURES       237

have

k = p

<; 1 for all / e B, and therefore

j.

Minorize this integral with the help of (b).) (Radernacher-KolmogorofT theorem.)

13.    Let X be a locally compact space, JLC a positive measure on X, and (un)n > 0 a sequence of
/x-integrable complex functions such that, as H runs through the set of finite subsets of

N, the set of numbers     ]T un(x) dp,(x) is bounded. Show that in these conditions the

J \n eH

oo

series ]T \un(x)\2 is convergent almost everywhere in X. (Observe that, for all

n = 0

t e I = [0, 1 ], the set of numbers     ^ uk(x)rk(t) d/x(x) is bounded above by a number
independent of n and t\ then use Problem 10(f) and the Lebesgue-Fubini theorem.)

14.    Let X be a compact space, ^ a positive measure on X, and (/„) an orthonormal
sequence in «£?c(X, /*) which is uniformly bounded in X.

(a)    Let (bn) be a sequence of real numbers such that ^ b% = +00. Show that it is

nsl

oo

not possible that £ \bn(g |/w)| < + oo for all continuous functions g on X. (Argue by

n= 1

contradiction : the functions un(x) = bnfn(x) would satisfy the conditions of Prob-
lem 13, by virtue of the Banach-Steinhaus theorem applied to the linear forms

0H+2W0I/-)   on   <f(X).   Hence f>fl2|/n(*)l2 < +°°   almost   everywhere  in  X.

neH                                                       it=l

Using EgorofT's theorem, the fact that the functions /„ are uniformly bounded and

/•                                                                                                                                                                                                                                00

l/nW|2 djj,(x) = 1, obtain a contradiction of the hypothesis V 6J = +00.)
J                                                                                       11=1

(b)    Let q be a real number such that 1 < q < 2. Show that it is not possible that

oo

2 \(9 \fn)\9 < -H oo for all continuous functions g on X. (If p = q\(q — 1), show that
if the assertion were false there would exist a sequence (bn) of real numbers > 0
such that 2 W ^ + °°» S W < + °° and S I W# !/«)! < + °° for a11 continuous

functions g.)

(c)   Deduce from (b) that there exists a continuous function g on X such that, for all

q satisfying 1 < q < 2, we havej) |(# |/B)|« = +00. (Use the principle of condensation

n

of singularities (Section 12.16, Problem 14).)

15. On the space R", let A be Lebesgue measure and let ||jc|| be a norm for which the unit
ball ||je|| <J 1 has measure 1. Let (#*)*>! be an infinite sequence of distinct points
in an integrable bounded set B such that A(B) = 1. For each integer m> 1, let
dm denote the smallest of the numbers ||#i — #/|| with 1 ^i<j<>m. Show that
lim inf m dm ^ c^ \ whereunless / is of the form k - 2"" with k integral and 0 < k < 2".
```